\begin{frame} \frametitle{The Definite Integral} \begin{block}{} % Let $f$ be a function defined on $[a,b]$. % \medskip % The \emph{definite integral of $f$ from $a$ to $b$} is \begin{talign} \alert{\int_{a}^{b} f(x)dx = \lim_{n\to \infty} \sum_{i = 1}^n f(x_i) \Delta x} \end{talign} provided that the limit exists, and has the same value for all possible choices of the \emph{sample points} \begin{center} $x_i$ from the interval $[a + (i-1)\Delta x,\; a + i\Delta x]$ \end{center} where $\Delta x = \frac{b-a}{n}$.\pause\medskip If the limit exists, we call $f$ \emph{integrable} on $[a,b]$. \end{block} \pause\smallskip The procedure of calculating an integral is called \emph{integration}. \pause\medskip Here $a$ is the \emph{lower limit} and $b$ is the \emph{upper limit} of integration. \pause \begin{block}{} The sum \quad \alert{$\sum_{i = 1}^n f(x_i) \Delta x$} \quad is called \emph{Riemann sum}. \end{block} \vspace{10cm} \end{frame}